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Is it true that for a set $\Omega$ with Lipschitz boundary the trace operator $T : H^1(\Omega) \to L^2(\partial \Omega)$ is compact? Can you please give a reference?

I found a theorem in Necas' Direct methods in the Theory of Elliptic Equations, which says that if $1<p<N, \ 1 \geq 1/q > 1/p-[1/(N-1)](p-1)/p$ then $W^{1,p}(\Omega)$ injects compactly into $L^q(\partial \Omega)$.

Unfortunately, a case which interests me is $p=q=2$ and $N=2$, which does not seem to fit in the above result. Can you please provide a reference for this case?

Servaes
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anonymus2345
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1 Answers1

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It does seem to fit in the above result, since $H^1 \subset W^{1,2-\varepsilon}$. It is better since if $p>2$, it is Holder continuous.

user1776247
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